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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Coordinate based co-localization</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">Co-localization is used to measure spatial overlap between two (or
more) different fluorescent labels, each having a separate emission
wavelength. Traditional co-localization analysis of conventional,
dual-color fluorescence microscopy images suffers from a limited spatial
resolution and chromatic errors <cite class="ltx_cite">[<a href="#bib.bib29" title="Spatial quantitative analysis of fluorescently labeled nuclear structures: problems, methods, pitfalls" class="ltx_ref">2</a>]</cite>. Coordinate
based co-localization (CBC) of dual-color super-resolution images
provides much better approach as it allows to determine co-localization
on a molecular level, it overcomes the dynamic range of a camera,
and it is not sensitive to cross talks <cite class="ltx_cite">[<a href="#bib.bib21" title="Coordinate-based colocalization analysis of single-molecule localization microscopy data" class="ltx_ref">1</a>]</cite>.</p>
</div>
<div id="Sx1.p2" class="ltx_para">
<p class="ltx_p">Calculation of the CBC value around a given molecule <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi17.png" width="23" height="18" alt="A_{i}">, according
to <cite class="ltx_cite">[<a href="#bib.bib21" title="Coordinate-based colocalization analysis of single-molecule localization microscopy data" class="ltx_ref">1</a>]</cite>, starts by determining two distributions
of distances</p>
<table id="Sx1.EGx1" class="ltx_equationgroup ltx_eqn_eqnarray">

<tr id="Sx1.Ex1" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi11.png" width="76" height="22" alt="\displaystyle D_{A_{i},A}\left(r\right)"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.Ex1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi8.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.Ex1.m3" class="ltx_Math" style="vertical-align:-19px" src="mi/mi14.png" width="163" height="47" alt="\displaystyle\frac{N_{A_{i},A}\left(r\right)}{N_{A_{i},A}\left(R_{\mathrm{max}%
}\right)}\frac{R_{\mathrm{max}}^{2}}{r^{2}}\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
<tr id="Sx1.Ex2" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.Ex2.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi13.png" width="76" height="22" alt="\displaystyle D_{A_{i},B}\left(r\right)"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.Ex2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi8.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.Ex2.m3" class="ltx_Math" style="vertical-align:-19px" src="mi/mi15.png" width="164" height="47" alt="\displaystyle\frac{N_{A_{i},B}\left(r\right)}{N_{A_{i},B}\left(R_{\mathrm{max}%
}\right)}\frac{R_{\mathrm{max}}^{2}}{r^{2}}\,."></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">Here <img id="Sx1.p2.m2" class="ltx_Math" style="vertical-align:-7px" src="mi/mi21.png" width="75" height="22" alt="N_{A_{i},A}\left(r\right)">, resp. <img id="Sx1.p2.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi22.png" width="76" height="22" alt="N_{A_{i},B}\left(r\right)">,
is the number of localized molecules in channel <img id="Sx1.p2.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi16.png" width="17" height="16" alt="A">, resp. <img id="Sx1.p2.m5" class="ltx_Math" style="vertical-align:-2px" src="mi/mi18.png" width="18" height="16" alt="B">,
within the distance <img id="Sx1.p2.m6" class="ltx_Math" style="vertical-align:-5px" src="mi/mi25.png" width="124" height="19" alt="r=0,\ldots,R_{\mathrm{max}}"> around <img id="Sx1.p2.m7" class="ltx_Math" style="vertical-align:-5px" src="mi/mi17.png" width="23" height="18" alt="A_{i}">.
These distributions are corrected for the area given by <img id="Sx1.p2.m8" class="ltx_Math" style="vertical-align:-2px" src="mi/mi26.png" width="13" height="12" alt="r"> and normalized
by the number of localizations within the largest observed distance
<img id="Sx1.p2.m9" class="ltx_Math" style="vertical-align:-5px" src="mi/mi23.png" width="44" height="18" alt="R_{\mathrm{max}}">.</p>
</div>
<div id="Sx1.p3" class="ltx_para">
<p class="ltx_p">Having these two distributions of distances, Sperman’s rank correlation
coefficient <img id="Sx1.p3.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi24.png" width="176" height="22" alt="S_{A_{i}}=\left\langle D_{A_{i},A}\,,\,D_{A_{i},B}\right\rangle">
is calculated. The co-localization value is determined for every single
molecule according to</p>
<table id="Sx1.Ex3" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex3.m1" class="ltx_Math" style="vertical-align:-19px" src="mi/mi9.png" width="218" height="46" alt="C_{A_{i}}=S_{A_{i}}\exp\left(-\frac{E_{A_{i},B}}{R_{\mathrm{max}}}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">where <img id="Sx1.p3.m2" class="ltx_Math" style="vertical-align:-7px" src="mi/mi20.png" width="50" height="21" alt="E_{A_{i},B}"> is a distance from localization <img id="Sx1.p3.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi17.png" width="23" height="18" alt="A_{i}"> to the
nearest neighbor localization in channel <img id="Sx1.p3.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi18.png" width="18" height="16" alt="B">.
</p>
</div>
<div id="Sx1.p4" class="ltx_para">
<p class="ltx_p">In ThunderSTORM, the input data for channels <img id="Sx1.p4.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi16.png" width="17" height="16" alt="A"> and <img id="Sx1.p4.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi18.png" width="18" height="16" alt="B"> are provided
as a table of results and a ground-truth table, respectively. Computed
values with the co-localization coefficient <img id="Sx1.p4.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi19.png" width="34" height="20" alt="C_{A_{i}}">, with the
distance to the nearest neighbor <img id="Sx1.p4.m4" class="ltx_Math" style="vertical-align:-7px" src="mi/mi20.png" width="50" height="21" alt="E_{A_{i},B}">, and with the number
of neighbors <img id="Sx1.p4.m5" class="ltx_Math" style="vertical-align:-7px" src="mi/mi22.png" width="76" height="22" alt="N_{A_{i},B}\left(r\right)"> within the radius <img id="Sx1.p4.m6" class="ltx_Math" style="vertical-align:-5px" src="mi/mi25.png" width="124" height="19" alt="r=0,\ldots,R_{\mathrm{max}}">,
are displayed in the table of results as new columns.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="results/Postprocessing.html" title="" class="ltx_ref">Post-processing methods</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="ImportExportPlugIn.html" title="" class="ltx_ref">Import and export of localization data</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib21" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">S. Malkusch, U. Endesfelder, J. Mondry, M. Gelléri, P. Verveer and Heilemann</span><span class="ltx_text ltx_bib_year">(2012-01)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Coordinate-based colocalization analysis of single-molecule localization microscopy data</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Histochemistry and Cell Biology</span> <span class="ltx_text ltx_bib_volume">137</span> (<span class="ltx_text ltx_bib_number">1</span>), <span class="ltx_text ltx_bib_pages"> pp. 1–10</span>.
</span>
<span class="ltx_bibblock">Note: <span class="ltx_text ltx_bib_note">Springer-Verlag</span>
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1007/s00418-011-0880-5" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Coordinate based co-localization" class="ltx_ref"><span class="ltx_text ltx_ref_title">Coordinate based co-localization</span></a>,
<a href="#Sx1.p2" title="Coordinate based co-localization" class="ltx_ref"><span class="ltx_text ltx_ref_title">Coordinate based co-localization</span></a>.
</span>
</li>
<li id="bib.bib29" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">O. Ronneberger, D. Baddeley, F. Scheipl, P.J. Verveer, H. Burkhardt, C. Cremer, L. Fahrmeir, T. Cremer and B. Joffe</span><span class="ltx_text ltx_bib_year">(2008-05)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Spatial quantitative analysis of fluorescently labeled nuclear structures: problems, methods, pitfalls</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Chromosome Research</span> <span class="ltx_text ltx_bib_volume">16</span> (<span class="ltx_text ltx_bib_number">3</span>), <span class="ltx_text ltx_bib_pages"> pp. 523–562</span>.
</span>
<span class="ltx_bibblock">Note: <span class="ltx_text ltx_bib_note">Springer Netherlands</span>
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1007/s10577-008-1236-4" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Coordinate based co-localization" class="ltx_ref"><span class="ltx_text ltx_ref_title">Coordinate based co-localization</span></a>.
</span>
</li>
</ul>
</div>
</div>
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